BAYESIAN BIOSTATISTICS - ITAMallman.rhon.itam.mx/~lnieto/index_archivos/Biostats1.pdf · BAYESIAN...

UNIVERSITY OF PERNAMBUCO

BAYESIAN BIOSTATISTICS

INSTRUCTORS:

LUIS E. NIETO-BARAJAS & LYN PIZANO-RODRIGUEZ

EMAILS: [email protected] & [email protected]

URL: http://allman.rhon.itam.mx/~lnieto

BAYESIAN BIOSTATISTICS, BY: LUIS E. NIETO & LYN PIZANO

II Encontro de Pós-Graduação e Pesquisa da Universidade de PernambucoRecife-PE, 8-10 de Outubro de 2008 2

BAYESIAN BIOSTATISTICS

Ø DEFINITIONS:

o Biostatistics (Wikipedia).

Is the application of statistics to a wide range of topics in biology. The

science of biostatistics encompasses the design of biological experiments,

especially in medicine and agriculture; the collection, summarization, and

analysis of data from those experiments; and the interpretation of, and

inference from, the results.

o Bayesian inference (Wikipedia).

Is statistical inference in which evidence or observations are used to update

or to newly infer the probability that a hypothesis may be true.

Ø OUTLINE:

1. Introduction

2. Exploratory Data Analysis

3. Probability Theory

4. Decision Theory

5. Bayesian inference

6. Priors

7. Clinical trial design

8. Hierarchical models

Appendix



Ø REFERENCES:

o Spiegelhalter, D. J., Abrams, K. R. and Myles, J. P. (2004). Bayesian

Approaches to Clinical Trials and Health-Care Evaluation. Wiley:

Chichester.

o Bernardo, J. M. (1981). Bioestadística: Una perspectiva Bayesiana. Vicens

Vives: Barcelona. (http://www.uv.es/bernardo/Bioestadistica.pdf)

o Bernardo, J. M. and Smith, A. F. M. (2000), Bayesian Theory. Wiley: New

York.

Ø SOFTWARE:

To understand the Bayesian concepts:

1) First Bayes (http://www.shef.ac.uk/~st1ao/1b.html)

To make statistical analyses:

2) WinBUGS (http://www.mrc-bsu.cam.ac.uk/bugs/)

3) Splus

4) R (http://www.r-project.org/)



1. Introduction

Ø The OBJECTIVE of Statistics, and in particular of Bayesian Statistics, is to

provide a methodology to adequately analyze the available information

(data analysis or descriptive statistics) and to decide in a reasonable way

the best way to proceed (decision theory or inferential statistics).

Ø DIAGRAM of Statistics:

Ø INFERENCE: It is the process to know population characteristics though a

subset of the population called sample. There are different ways to make

inference:

Assumption \ Approach Classic Bayesian

Parametric √√√ √√

Non parametric √√ √

Population

Sample

Sampling Inference

DDaattaa aannaallyyss iiss

DDeecciiss iioonn mmaakkiinngg



Ø Some basic definitions:

o Element or individual: Object (person, item, animal, planta, etc.) whose

properties are to be analyzed.

o Population: A Collection of individuals or objects.

o Sample: A subset of the population.

o Parameter: A numerical value summarizing all the data in an entire

population.

o Statistic: A numerical value summarizing the sample data.

Ø VARIABLE: Characteristic or feature to be measure in an individual.

Ø ¿How to get a representative sample?

o Through random selection with a probability scheme. (Randomized trials !)

o The selection is made with replacement or without replacement if the

population is large ⇒ to induce independence.

Types of variables

Numeric Categorical

Continuous Discrete Ordinal Nominal



2. Exploratory data analysis

Ø Assume that the collecting of data has been made. Let X1,X2,...,Xn be a

sample of size n of observations from the variable of interest X, where each

Xi represents the characteristic of interest for individual i.

Ø Exploratory techniques are divided in two:

1) Graphic techniques, and

2) Descriptive measures.

Ø Study: Measure the survival times of 100 terminal cancer patients who

were given supplemental ascorbate (Vitamin C) as part of their routine

management and 1000 matched controls (similar patients who have

received the same treatment except for the ascorbate).

Objective: Determine whether supplemental ascorbate prolongs the

survival times of patients with terminal human cancer.

Variables: Cancer type, Sex, Age (years), Survival times (days) for both

cases and controls.

Graphic techniques:

Ø CATEGORICAL VARIABLES:

o Barplot: displays the frequencies or relative frequencies of each category.

o Piechart: Displays the relative frequencies of a categorical variable as the

size of a piece of pie.



0.530.47

Female Male1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

0.5

Sex

Barplot for Sex

Stomach

Bro

nchu

s

Colon

Rectum

Ovary

Brea

st

Bla

dder

Kidney Gall_Bladder

Esophagus

Reticulum

Prostate Uterus

Brain

PancreasCLL

Piechart for Type (in percentage)

13171776117822221131



Ø NUMERIC VARIABLES:

o Stem and leaf plots: Shows the form of the distribution of observed values

in a vertical position.

o Frequencies distribution: Contains frequencies and relative frequencies,

absolute and cumulative.

o Histogram: Graphical representation of the relative frequencies.

Stem and leaf plot for Age Decimal point is 1 place to the right of the colon

3 : 89

4 : 344

4 : 688999

5 : 01122333344

5 : 5566666777788999

6 : 00122233344

6 : 556667778888888999999

7 : 000000111233444444

7 : 5566677779

8 : 0

8 :

9 : 3

Frequencies distribution for Age

class lower.l upper.l freq rel.freq cum.freq rel.cum.freq [1,] 35 40 2 0.02 2 0.02 [2,] 40 45 3 0.03 5 0.05 [3,] 45 50 7 0.07 12 0.12 [4,] 50 55 12 0.12 24 0.24 [5,] 55 60 16 0.16 40 0.40 [6,] 60 65 11 0.11 51 0.51 [7,] 65 70 25 0.25 76 0.76 [8,] 70 75 14 0.14 90 0.90 [9,] 75 80 9 0.09 99 0.99 [10,] 80 85 0 0.00 99 0.99 [11,] 85 90 0 0.00 99 0.99 [12,] 90 95 1 0.01 100 1



40 50 60 70 80 90

0.0

0.01

0.02

0.03

0.04

0.05

Age

Histogram for Age

o In the case of survival data, because of the presence of censored

observations, it is better to produce Kaplan-Meier plots: This graph

consists of plotting the empirical probability of dying (or presenting the

event of interest) after time “t”.

Time (days)

Sur

viva

l pro

babi

lity

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

CasesControls

Kaplan-Meir plots for Cases and Controls



Numerical descriptive measures:

Ø Numerical measures could be either of central tendency, position, or

dispersion.

Ø CENTRAL TENDENCY MEASURES: These measures locate the central part of a

the values of a variable. The three most important are:

o Mean: is the arithmetic average of the observations.

∑=

=n

1iiX

n1

X = Sample mean

The mean is not a good central measure when the distribution of the

data is skewed.

o Mediana: is the observation that lies just at the middle of a dataset alter

being ordered.

l = n×0.5+0.5 = position of the median

m = X(l) = median (observation X that lies at position l

after ordering the data).

The median is a good indicator of central tendency when the

distribution of the data is skewed.

o Mode: is the observation that occurs the most frequently in a dataset. If

this value is unique we say that the frequencies distribution is unimodal.



Ø POSITION OR LOCATION MEASURES: These are called quantiles or

percentiles. For p∈(0,1), the pth percentile is the observation that divides

the dataset such that p×100% of the observations are smaller and (1-

p)×100% are larger. The most common percentiles are:

o Quartiles: are observations that divide a dataset in 4 parts of equal

number of observations.

Q1 = X(n×0.25+0.5) = First quartile

Q2 = X(n×0.50+0.5) = Second quartile

Q3 = X(n×0.75+0.5) = Third quartile

Ø DISPERSION MEASURES: These are measures of the variability

(concentration, dispersion) of a dataset. The most common measures are:

o Range: is the simplest measure and indicates the spread between the

smallest and the largest observations.

R = Maximum – minimum = range

o Interquartile range: is the distance between the first and the third

quartiles.

ICR = Q3 – Q1 = interquartile range

o Variance: is the average of squared deviations of each obervation to the

mean.

( )

−

−=−

−= ∑∑

==

2n

1ii

n

1i

2i

2 XnX1n

1XX

1n1

S = sample variance



The square root of the variance is called standard deviation, i.e.,

2SS = = sample standard deviation

o Variation coefficient: measures the relative dispersion of a dataset with

respect to the location.

XS

cv = = sample variation coefficient

This measure is useful to compare the variability of two datasets

because it does not depend on the measuring scale.

Descriptive measures for Age

Min. 1st Qu. Median Mean 3rd Qu. Max. 38 56 65 63.2 70 93

Descriptive measures for Survival times

Variable n events mean se(mean) median Cases 100 92 781 112 331 Controls 100 100 360 32.2 269



Ø BOXPLOT DIAGRAMS: The boxplot summarizes the most important

descriptive measures. It also allows us to assess symmetry and the presence

of outliers. This diagram is also useful to compare different variables. 0

1000

2000

3000

4000

Cases Controls



3. Probability Theory

Ø Probability theory acts as a bridge between descriptive statistics and

inferential statistics.

Ø Informally: Probability is a quantification or measure of the uncertainty

associated to the occurrence of an event.

Ø Formally: Probability is a function that satisfies 3 axioms:

1. ( ) 0AP ≥ for all event A

2. ( ) 1P =Ω , where Ω contains all possibilities

3. ( ) ( ) ( )BPAPBAP +=∪ , if ∅=∩ BA

Ø Although there is only one mathematical definition of probability, there are

several ways of assigning a probability: classic, frequentist and subjective.

For example, if somebody says that the probability of a coin coming up

heads is ½, how did he get this number?

Data analysis (Descriptive S.)

Decision theory (Inferential S.)

Probability



Ø Something important about probabilities is that the quantification of

uncertainty is subject to change according to the conditions or the

knowledge we have about the event ⇒ conditional probabilities.

o Example: Consider the experiment of tossing two fair coins, and let A, B

and C three events such that

A=two heads

B=the first coin is head

C=at least one of the coins is head

P(A)=1/4

P(A given that we know B)=1/2

P(A given that we know C)=1/3

Ø CONDITIONAL PROBABILITY: Once we know that event B has occurred we

are interested in the probability of A. This is obtained by,

( ) ( )( )BP

BAPBAP

∩= , if ( ) 0BP ≠

o Comment: Broadly speaking, all probabilities are conditional probabilities:

( )HAP .

Ø From the definition of conditional probability we can derive two important

results: the marginalization rule and the Bayes’ theorem.

o Result 1: Marginalization rule.

( ) ( ) ( ) ( ) ( )cc BPBAPBPBAPAP += ,

where =cB not B.



o Example: Prognosis.

We wish to determine the probability of survival (up to a specified point in

time) following a particular cancer diagnosis, given that it depends on the

stage of disease at diagnosis among other factors. Let

A=surviving

B=cancer was diagnosed at an early stage

Bc= cancer was NOT diagnosed at an early stage

Computing P(A) directly may be difficult, but we can obtain it by using the

marginalization rule.

Suppose patients with early stage disease have good prognosis, say

( ) 80.0BAP = , but for late stage it is poor, say ( ) 20.0BAP c = . We also

know that the majority of all diagnoses are early stage, that is , ( ) 90.0BP = ,

and therefore ( ) 10.0BP c = . Then the marginal probability of surviving is:

( ) 74.010.020.090.080.0AP =×+×=

o Result 2: Bayes Theorem.

( ) ( )( ) ( )BPABP ×=AP

BAP

This theorem tells us formally the learning process: ( ) ( )ABPBP → .

o Example: Prognosis (cont…)

The probability that the disease was diagnosed at an early stage can be

updated if we know that the patient has survived. A priori we knew that

P(B)=0.90. Now suppose that we find out that the patient survived, that is,

we know A. Then a revised probability of an early diagnosis is:



( ) 97.090.074.080.0ABP =×=

Ø ODDS AND LOG-ODDS: An alternative way of reporting a probability.

Instead of quantifying the uncertainty in the [0,1] scale, we can do it in the

[0,∞) scale:

p1p

O−

=

and

O1Op+

= .

The natural logarithm of the odds is called logit,

( ) ( )∞∞−∈

−

= ,p1

plogpitlog .

o Example: a probability of 0.20 (20% chance) corresponds to odds of

O=0.20/0.80=0.25 or, in betting language, “4 to 1 against”. Conversely,

betting odds of “7 to 4 against” correspond to O=4/7 or a probability of

p=4/11=0.36.

o Bayes Theorem for odds: The learning mechanism given by the Bayes

Theorem can also be written in terms of odds:

( )( )

( )( )

( )( )cc BPBP

ABPABP

×= cBAP

BAP.

Or equivalently,

( )( )

( )( )

( )( )BP1

BPBAP1

BAPABP1

ABP−

×−

=−

.



This form of the Bayes Theorem allows us to update ( )BP into ( )ABP

without calculating ( )AP .

o Example: Prognosis (cont…)

The initial odds for an early stage diagnosis are:

( ) ( ) 910.0/90.0BP/BP c == ,

the ratio ( ) ( ) 420.0/80.0BAP/BAP c == , therefore the updated odds are

( ) ( ) 3694ABP/ABP c =×= .

Ø BAYESIAN THEORY is based on the subjective interpretation of probability

and has its roots in Bayes Theorem.



4. Decision theory

Ø Statistical Inference is a way of making decisions. Classical methods of

inference ignore important aspects of the decision-making process;

however, Bayesian methods of inference do take them into account.

Ø What is a decision problem? We face a decision problem when we have to

select from two or more ways of proceeding.

Ø MAKING DECISIONS is a fundamental aspect in the life of a professional

person, for instance, a physician must make decisions constantly in an

environment with uncertainty, decisions about the best treatment for a

patient, etc.

Ø DECISION THEORY proposes a method of making decisions based on some

basic principles about the coherent election between alternative options.

Ø ELEMENTS OF A DECISION PROBLEM under uncertainty:

A decision problem is defined by the quadruplet (D, E, C, ≤), where:

q D : Space of decisions. Set of possible alternatives, it has to be exhaustive

(contains all possibilities) and exclusive (electing one element in D

excludes the election of any other).

D = d1,d2,...,dk.



q E : Space of uncertain events. Contains uncertain events relevant to the

decision problem.

Ei = Ei1,Ei2,...,Eimi., i=1,2,…,k. Ei = E i1, E i2,K, E im i ,i = 1,2,K, k

q C : Space of consequences. Set of all possible consequences and describes

the consequences of electing a decision.

C = c1,c2,...,ck.

q ≤ : Preference relation among different options. Is defined in such a way

that d1≤d2 if d2 is preferred over d1.

• REMARK: For the moment we will consider discrete spaces (decisions,

events and consequences), although the theory is also applied to continuous

spaces.

Ø DECISION TREE (under uncertainty):

d1

di

dk

c11 c12

c1m1

There is not full information about the consequences of making a decision

E11

E12

E1m1

Ei1

Ek1

Ei2

Ek2

Eimi

Ekmk

ci1

ck1

ci2

ck2

cimi

ckmk



Ø Example: Decision problem.

A physician needs to decide whether to carry out surgery on a person he

believes has a malignant tumor or to treat with chemotherapy. If the patient

has a benignant tumor, the life expectancy is 20 years. If he has a

malignant tumor, undergoes surgery, and survives, he is given 10 years of

life; whereas if he has a malignant tumor and does not undergo surgery, he

is only given 2 years of life.

D = d1, d2, where d1 = surgery, d2 = therapy

E = E11, E12, E13, E21, E22, where E11 = survival / tumor, E12 = survival /

no tumor, E13 = dead, E21 = tumor, E22 = no tumor

C = c11, c12, c13, c21, c22, where c11=10, c12=20, c13=0, c21=2, c22=20

Decision node Uncertainty (random) node

Surgery

Therapy

Surv

Dead

M.Tum

B.Tum

M.Tum

B.Tum

10 yrs.

20 yrs.

0 yrs.

2 yrs.

20 yrs.



Ø In practice, most decision problems have a much more complex structure.

For instance, one may have to decide whether or not to carry out an

experiment, and if one does the experiment, make another decision

according to the result of the experiment. (Sequential decision problems).

Ø Frequently, the set of uncertain events is the same for all decisions, that is,

Ei = E i1, E i2,K, E im i = E1, E 2,K, Em = E , for all i. In this case, the

problem can be represented as:

E1 ... Ej ... Em

d1 c11 ... c1j ... c1m

M M M M

di ci1 ... cij ... cim

M M M M

dk ck1 ... ckj ... ckm

Ø The OBJECTIVE of a decision problem under uncertainty is then to make the

best decision di from the set D without knowing which of the events Eij

from Ei will occur.

Ø Although the events that form each Ei are uncertain, in the sense that we do

not know which of them will occur, in general, we have an idea of the

probability of each of them. For instance,



25 years

Ø Sometimes it is difficult to order our preferences among all possible

different consequences. It might be simpler to assign a utility measure to

each of the consequences and then order them according to their utility.

Ø QUANTIFICATION of uncertain events and of consequences.

q The information that the decision maker has about the possible occurrence

of the events can be quantified through a probability function on the space

E.

live 10 yrs. more

die in 1 month

reach 90 yrs.

Which is more probable?

Consequences Earn much money & have little available

time

Earn little money & have much available

time

Earn regular money & have regular available

time



q In the same way, it is possible to quantify the preferences of the decision

maker among different consequences through a utility function in such a

way that 'j'iij cc ≤ ⇔ ( ) ( )'j'iij cucu ≤ .

Ø Alternatively, it is possible to represent the decision tree as follows:

Ø How to make the best decision?

If in some way we were able to make the uncertainty disappear, we could

order our preferences according to the utility of each decision. Then the

best decision would be the one that has the maximum utility.

d1

di

dk

u(c11)

u(c12)

u(c1m1)

P(E11|d1)

P(E12|d1)

P(E1m1|d1)

P(Ei1|di)

P(Ek1|dk)

P(Ei2|di)

P(Ek2|dk)

P(Eimi|di)

P(Ekmk|dk)

u(ci1)

u(ck1)

u(ci2)

u(ck2)

u(cimi)

u(ckmk)

Uncertainty Decider Go away



Ø STRATEGIES: There are 4 strategies or criteria proposed in the literature to

disappear the uncertainty and make decisions: Optimistic, pessimistic or

minimax, conditional or most probable, and expected utility.

q Whichever strategy one takes, the best option is the one that maximizes the

tree “without uncertainty.”

Ø AXIOMS OF COHERENCE. These are a series of principles that establish the

conditions for making coherent decisions and that clarify the possible

ambiguity in the process of making a decision. There are four axioms of

coherence:

1. COMPARABILITY. This axiom establishes that we should at least be able

to express preferences between two different options.

2. TRANSITIVITY. This axiom establishes that preferences must be

transitive to avoid contradictions.

3. SUBSTITUTION AND DOMINATION. This axiom establishes that there are

equivalent options and there are also options dominated by others.

4. REFERENCE EVENTS. This axiom establishes that to be able to make

reasonable decisions, it is necessary to measure the information and the

preferences of the decision maker in a quantitative form.

Ø IMPLICATIONS: As a consequence of the axioms, if we want to make

coherent decisions, the way of making a decision is as follows:



1) Assign a utility u(c) for all c in C.

2) Assign a probability P(E) for all E in E.

3) Select the (optimal) option that maximizes the expected utility.

o Theorem: Bayesian decision criteria.

The expected utility of the option di = ijij m,,1j,Ec K= is defined as:

( ) ( ) ( )∑=

=im

1jiijiji dEPcudu .

Then the optimal decision is d∗ such that ( ) ( )ii

* dumaxdu = .

Ø Example. Decision problem (cont…).

Assume that the prior believes of the physician are that a patient survives

the surgery 90% of the times and 60% of the tumors are malignant tumor.

We consider that undertaking a surgery is independent of the condition of

the tumor. Furthermore, assuming that the utility is proportional to the

years of life, then the decision problem is re-written as

Surgery

Therapy

(0.9) Surv

(0.1) Dead

(0.6) M.Tum

(0.4) B.Tum

(0.6) M.Tum

(0.4) B.Tum

10 yrs.

20 yrs.

0 yrs.

2 yrs.

20 yrs.



Then, the expected utility of each option becomes

( ) ( )( ) ( )( ) ( ) 6.121.004.09.0206.09.010du 1 =++= , and

( ) ( ) ( ) 2.94.0206.02du 2 =+=

Therefore, the option that maximizes the expected utility is d1, that is, the

optimal decision is to carry out surgery.

Ø FINAL COMMENT:

The more we know about the uncertain events,

the better the decision made is.

How do we reduce uncertainty about E?

Obtaining additional information (Z) about the events E’s. We then update

our knowledge by using the Bayes Theorem, that is,

( )EP ( )ZEP

In this case we have two situations:

1) Initial situation (a-priori):

( )ijEP , ( )ijcu , ( ) ( ) ( )∑=j

ijiji EPcudu

2) Final situation (a-posteriori):

( )ZEP ij , ( )ijcu , ( ) ( ) ( )∑=j

ijiji ZEPcuZdu

In any case, the option that maximizes the expected utility is the optimal

decision.

Bayes Theo.

Initial expected

utility

Final expected

utility



5. Bayesian inference

Ø Let X = random variable of interest

(e.g. response to a drug or the survival time of patients).

Ø The behavior of X depends, in a parametric world, on the value of some

unknown quantities θ called parameters.

( )θ⇒ xf

denotes the density function of X that depends on θ.

Ø INFERENCE PROBLEM. Let F ( ) Θ∈θθ= ,xf be a parametric family indexed

by the parameter θ∈Θ. Let X1,...,Xn be a random sample (r. s.) of

observations from f(x|θ) ∈F. The inference problem consists of estimating

the real value of the parameter θ.

o In a Bayesian perspective, the inference problem can be seen as a decision

problem with the following elements:

D = space of decisions (in point estimation, D = Θ)

E = Θ (parameter space)

C = ( ) Θ∈θ∈θ ,d:,d D

≤ : will be represented by a utility function or a loss function.

Ø The sample gives additional information about the uncertain events θ ∈ Θ.

The problem consists of how to update the information.



Ø If the coherence axioms are accepted, the decision maker is capable of

quantifying his or her knowledge about the uncertain events through a

probability function. We then define,

( )θf the prior distribution (or a-priori). Quantifies the initial

knowledge about θ.

( )θxf sample information generating process. Gives additional

information about θ.

( )θxf the likelihood function. Contains all information about θ given

by the sample ( )n1 X,XX K= . ( ) ( )∏=

θ=θn

1iixfxf

q All this information about θ is combined to obtain a final knowledge or a-

posteriori after having observed the sample. The way to do it is by means

of Bayes Theorem:

( ) ( ) ( )( )xf

fxfxf

θθ=θ ,

where ( ) ( ) ( )∫Θ

θθθ= dfxfxf or ( ) ( )∑θ

θθ fxf .

As ( )xf θ is a function of θ, then we can write

Finally,

( )xf θ the posterior distribution (or a-posteriori). Summarizes all

available knowledge about θ (prior + sample).

( ) ( ) ( )θθ∝θ fxfxf



Ø Example: Tumor response.

X=tumor response under a therapy

=otherwise 0,

response positive if ,1x

( ) ( )θ=θ⇒ xBerxf ,

where θ=probability of response.

The prior believes of the experts are that the probability of response (θ) for

this new therapy is well represented by

( ) ( )3,3Betaf θ=θ

theta

0.0 0.2 0.4 0.6 0.8 1.0

01

23 Prior

After testing the therapy on n=10 patients, only 2 of them responded

positively, which give us a likelihood

( ) ( )82 1xf θ−θ=θ



theta

0.0 0.2 0.4 0.6 0.8 1.0

01

23 Prior

Likelihood

Combining the prior with the additional information given by the

likelihood, we get a posterior knowledge about θ given by

( ) ( )11,5Betaxf θ=θ

theta

0.0 0.2 0.4 0.6 0.8 1.0

01

23 Prior

LikelihoodPosterior



Ø REMARK: As θ is a random quantity, since we are uncertain about the true

value of θ, the density function ( )θxf that generates relevant information

about θ is actually a conditional density. Moreover, as θ is unknown, f(x|θ)

can not be used to describe the behavior of the r.v. X.

Ø PREDICTIVE DISTRIBUTION: The preditive distribution is the marginal

density function f(x) that allows us to determine which values of the

random variable are more probable.

1) Prior predictive distribution. Using the prior ( )θf and marginalizing

( ) ( ) ( )∫ θθθ= dfxfxf or ( ) ( ) ( )∑θ

θθ= fxfxf

2) Posterior predictive distribution. Using the posterior ( )xf θ and

marginalizing

( ) ( ) ( )∫ θθθ= dxfxfxxf FF or ( ) ( ) ( )∑θ

θθ= xfxfxxf FF

Ø Example: Tumor response (cont…).

Our idea is to determine the probability of response for a set of m=10 new

patients, say ( )θ∼= ∑=

,10BinXY10

1ii , ← θ unknown.

The posterior knowledge we have about θ is that ( ) ( )11,5Betaxf θ=θ . One

alternative to determine the value of θ is to take the average (posterior

mean), that is,

( ) 31.0xEˆ =θ=θ ⇒ ( )31.0,10BinY ≈ ← plug-in



However, this procedure does not take into account the uncertainty around

θ. So the correct answer will be given by the posterior predictive

distribution which takes the form

( ) ( )11,5,10BeBinxyf = ← Beta−Binomial 0.

00.

050.

150.

25

Y

Binomial distribution

0.0

0.05

0.15

0.25

Y

Beta-Binomial distribution

Beta-BinomialBinomial

0.0

0.05

0.10

0.15

0.20

0.25

Y

Precitive distribution



Ø MODEL COMPARISON. ( )111 ,xf:M θ vs. ( )222 ,xf:M θ

We can naturally solve this problem by considering a decision problem but

alternatively, we can compute a Bayes factor (likelihood ratio)

( )( )xfxfB

2

1= ,

where ( ) ( ) ( ) ( )∫ θθπθ= jjjjj dxfxf , j=1,2.

If B is large (>10) ⇒ data supports M1

If B is small (<1/10) ⇒ data supports M2

Ø SUMMARY: Bayesian analysis.

( )θxf and ( )θf are probability distributions that define a joint model

( )θ,xf , which implies a posterior ( )xf θ and a predictive (marginal) ( )xf .

o Posterior probabilities are over parameter space Θ, e. g.

§ ( )x15.0P >θ

§ ( )xP 21 δ+θ>θ , etc.



6. Priors

Ø There exist several classes of prior distributions. In terms of the amount of

information they carry, we classify them as informative and

noninformative.

Ø INFORMATIVE PRIOR DISTRIBUTIONS: These are prior distributions that

contain relevant information about the occurrence of the uncertain events

θ. There are two kinds:

o Subjective prior: probability model reflecting (personal) judgement

about uncertain events (parameter values).

o Historical prior: (from related studies) judgement about uncertain

events (parameter values) informed by related earlier studies. We can

achieve a historical prior by:

§ Discount with inflated variance, or

§ Use only a fraction of the data set.

Ø Example: Amount of tyrosine.

The consequences of certain treatment can be determined by the amount of

tyrosine (θ) in the urine. The prior information about this quantity in

patients shows that it is around 39mg./24hrs. and that the percentage of

times this quantity exceeds 49mg./24hrs. is 25%.

According to this information, “it can be implied” that the normal

distribution models “reasonably well” this behavior, so



θ ∼ ( )2,N τµ ,

where µ=E(θ)=mean and τ2=Var(θ)=variance. Moreover,

How?

( ) 25.03949

ZP49P =

τ−

>=>θ ⇒ τ−= 3949Z 25.0 ,

as Z0.25 = 0.675 (from tables) ⇒ τ = 100.675

Therefore, θ ∼ N(39, 219.47).

theta

Den

sity

0 20 40 60 80

0.0

0.00

50.

010

0.01

50.

020

0.02

5

Informative prior for Tyrosine

Amount of tyrosine (θ) around 39 µ=39

P(θ > 49) = 0.25 (given µ=39)

τ=14.81



Ø Example: Amount of tyrosine (cont...)

Suppose now that there exist only 3 possible values (categories) for the

amount of tyrosine: θ1 = low, θ2 = medium, & θ3 = high. Assume even

further that θ2 is three times as frequent as θ1 and that θ3 is twice as

frequent as θ1. We can specify the prior distribution for the amount of

tyrosine by,

letting pi=P(θi), i =1,2,3. Then,

p2=3p1 and p3=2p1. Moreover, 1ppp 321 =++

⇒ p1+ 3p1+ 2p1=1 ⇔ 6p1=1 ∴ p1=1/6, p2=1/2 and p3=1/3

Ø NONINFORMATIVE PRIOR DISTRIBUTIONS: These are prior distributions that

do not give us any relevant information about the occurrence of the

uncertain events θ. There are several criteria to define a noninformative

prior:

1) Principle of insufficient reasoning: According to this principle, in the

absence of evidence against, all possibilities have the same prior

probability.

⇒ Uniform priors

2) Invariant prior distribution: Jeffreys (1946) proposed a noninformative

prior distribution that is invariant under re-parameterizations.

2/1)(Idet)( θ∝θπ , θ∈Θ,

where ( )

θ∂θ∂θ∂

−=θ θ ' Xflog

E)(I2

|X is Fisher’s information matrix.



o Example: Let X be a r.v. with conditional distribution given θ,

( ) ( )θ=θ xBerxf , i.e., ( ) ( ) )x(I1xf 1,0x1x −θ−θ=θ , θ∈(0,1). Then,

( )2/1 ,2/1Beta)( θ=θπ ← Jeffreys prior

( )1 ,0Uniform)( θ=θπ ← Insufficient reasoning prior

theta

Den

sity

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

JeffreysUniform

Noninformative priors

Ø COMMENTS on noninformative priors:

o Useful for data analysis

o Impractical for design problems: need to consider inference before

recording data

o Solution: Consider two different priors:

§ design prior (optimistic informative) vs.

§ analysis prior (skeptic, vague)

optimistic = investigator, drug developer

skeptic = regulator, decision maker



Ø CONJUGATE PRIORS: Prior distribution such that the posterior ( )xθπ and the

prior ( )θπ belong to the same family (i.e., have the same form but with

updated parameters).

o Example: Let X1,X2,...,Xn be a r.s. from ( ) ( )θ=θ xBerxf .

Prior: ( ) ( ) ( ) )(I1)b()a()ba(

b,aBeta )1,0(1b1a θθ−θ

ΓΓ+Γ

=θ=θπ −−

Likelihood: ( ) ( ) ( )∏=

−∑∑ θ−θ=θn

1ii1,0

xnxxI1xf ii

Posterior: ( ) ( ) )(I1)b()a()ba()b,a(Betaxf )1,0(

1b1a

11

1111

11 θθ−θΓΓ+Γ=θ=θ −−

where, ∑+= i1 xaa and ∑−+= i1 xnbb .

o Example: Let X1,X2,...,Xn be a r.s. from ( ) ( )2,xNxf σµ=θ .

( ) ( )200 ,N σµµ=µπ is the conjugate prior for µ, and

( ) ( )0022 b,aGamma −− σ=σπ is the conjugate prior for σ−2.

if ∞→σ20 then ( ) .cte∝µπ ← improper noninformative prior

if 0a 0 ≈ and 0b 0 ≈ then ( ) ( )001.0,001.0Gamma 22 −− σ=σπ ←vague prior

o More examples of conjugate families can be found in the list of formulas.

http://www.uv.es/~bernardo/FormulBT.pdf



7. Clinical trial design

Ø “The main objective of almost all trials on human subjects is (or should be)

a decision concerning the treatment of patients in the future”.

Ø The most commonly performed clinical trials evaluate new drugs, medical

devices (like a new catheter), biologics, psychological therapies, or other

interventions.

Ø Clinical trials may be required before the national regulatory authority will

approve marketing of the drug or device, or a new dose of the drug, for use

on patients.

Ø For drug development trials (pharmaceutical industry) we have several

phases:

1) Phase I study: deal with identifying a safe dose (maximum tolerable

dose without toxicities), usually on healthy volunteers.

2) Phase II study: concerned with finding an effective dose.

3) Phase III study: are intended to prove treatment benefit over an

appropriate control.

4) Phase IV study: monitor the use and possible side-effects of a drug in

routine use.

Ø The objective of the trial is usually specified as statistical hypothesis of

clinically meaningful events, that is, in terms of the parameters of the

model.

For instance:



H0: treatment equivalence

H1: superiority of new treatment

(H2: inferiority of new treatment)

Ø Example: Lung cancer trial.

The physicians would be willing to use routinely the new treatment only if

it confers at least 13.5% improvement in two year survival (from a baseline

of 15%), and unwilling if less than 11% improvement.

Thus, if T= time to dead, and θ=P(T>2), then

H0: [ ]285.0,26.0∈θ ← range of equivalence

H1: θ > 0.285

H2: θ < 0.26

Ø Example: Tumor response.

Stopping criteria based on posterior probabilities.

Let 1,0Yi ∈ be the tumor response under new therapy for patient i.

( )1YP i ==θ .

Suppose that the standard of care is %150 =θ , and that the range of

equivalence is [ ]%20 , %10 , therefore

H0: [ ]20.0,10.0∈θ

H1: θ > 0.20

H2: θ < 0.10

Let n21 y,,y,y K be the response for n patients then we need to evaluate

two situations:



1) Stop and recommend the experimental therapy if

( ) ( ) 1n1n11 y,,y2.0Py,,yHP π>>θ=∈θ KK

2) Stop and abandon the experimental therapy if

( ) ( ) 2n1n12 y,y1.0Py,yHP π><θ=∈θ KK or maxnn ≥

1π and 2π are set to be close to one and are called design parameters.

These can be tuned to achieve desired frequentist properties !!.

Ø FREQUENTIST OPERATING CHARACTERISTICS of a design.

Type –I error:

( )0recommend and stopP θ=α .

Comments:

o Typically Analytically intractable

o Require (independent) Monte Carlo simulation

Ø Example: How to compute type−I error.

1. Fix 0θ=θ

2. Simulate a possible history of the trial

a. Simulate ( )θ∼ ii yfy , n,,1i K=

b. Evaluate posterior probabilities; e.g. ( )n1 y,,y2.0P K>θ

c. Evaluate stopping rules if applicable

d. Upon completion of the trial, record the final decision

3. Repeat the trial simulation M times

4. Record the number of trials MR that end with the final decision of

rejecting H0 and report

MMR≈α



Ø DECISION THEORETIC DESIGN.

This is a design based on entirely on the decision theory framework, that is,

we need a space of decisions, uncertain events, consequences and

quantifications: utility and probability model.

o Space of decisions: Dd ∈ , where for example,

Ex 1: choice of the nest dose, 1it zd +=

Ex 2: stopping decision, 2,1,0d t ∈ ,

where 0=stop and abandon, 1=phase III, 2=continue accrual

o Probability model: Quantification of the uncertainty of all unknown

quantities: parameters θ, historical data y0, data y and latent data y~ .

Ex: Dose/Response problem, ( ) ( )( )2iiii ,z,gyNz,yf σθ=θ , n,,1i K= and

prior ( )2,f σθ , with mean response ( )z,g θ at dose z.



o Utility function: ( )θ,du worth of a decision d under uncertain event θ.

Ex 1: Precision of the dose effect θ, i.e., ( )tzVar1 θ

Ex 2: Sampling cost + reward of success,

( ) ( )( )

=θ+⋅

=⋅+⋅

=

=θ

+ 2d if ,,dUnc

1d if ,SPCmc

0d if ,0

,dU

t*

1t

t

t

t

where m=phase III sample size

S=significant phase II trial

n=cohort size

*1td + =optimal decision at time t+1



8. Hierarchical models

Ø The Bayesian hierarchical models simplify the simultaneous estimation of

several parameters θi of the same type with two objectives:

1) Borrow strength to improve precision in the estimation of parameters

2) Allow introduce uncertainty in the estimations

Ø In general, we can borrow strength across multiple related

§ Studies

§ Subpopulations

§ Current and historical studies

§ Diseases

§ Etc…

Ø Consider multiple studies (sub-populations):

( )1n1111 y,,yy K= , ( )

2n2212 y,,yy K= , … , ( )kkn1kk y,,yy K=

The hierarchical model can be summarized in the following diagram:

1Y 2Y kY

1θ 2θ kθ

ϕ Hyper-parameter

parameter

observations



Ø FORMALLY, the hierarchical model can be specified as:

1) Parameters and sub-models for each study:

( )11yf θ , ( )22yf θ , …

2) Borrow strength across studies by combining parameters at the prior

level:

( )ϕθ1f , ( )ϕθ2f , …

together with

( )ϕf

§ Note: The prior on θj could include regression on study specific

covariates:

( )ϕθ ,zf jj

Ø There are two alternatives to the hierarchical models:

1. Weaker dependence. Assuming independent studies: separate ϕj’s

( )jjj f ϕθ∼θ and ( )jf ϕ

2. Stronger dependence. Assuming exchangeable patients (pooling):

common θ

( )θ∼ ii yfy

o Remark: The hierarchical model is a compromise between 1 and 2.

Ø The main application of hierarchical models has been to carry out

META−ANALYSES (quantitative synthesis of multiple studies).

Exchangeability



Ø Example: EFM: meta−analysis of trials with rare events. (Spiegelhalter et

al. 2004, p. 275)

EFM: Electronic fetal heart rate monitoring in labor.

Aim: Early detection of altered heart rate pattern and hence a potential

benefit in perinatal mortality.

Number of studies: 9 randomized trials

Outcome: Perinatal mortality measured as odds ratio in deaths per 1000

births (comparing EFM vs. control).

Statistical models:

Let ( )ORlogj =θ and Yj the observed OR then

( )2jj ,Ny σθ∼

a) Approximate normal likelihood + fixed effects:

( )2j ,N τµ∼θ

b) Approximate normal likelihood + random effects:

( )2j ,N τµ∼θ , Uniform∼µ , Uniform∼τ

Let tkr the observed deaths in the treatment group, and

ckr the observed deaths in the control group

tkn and ckn the total number of patients in each group

Then

( )tktktk p,nBinr ∼ and ( )ckckck p,nBinr ∼

with

( ) kktkpitlog θ+φ= and ( ) kckpitlog φ=



c) Binomial likelihood + random effects (uniform risks)

( )2j ,N τµ∼θ , Uniformp ck ∼

d) Binomial likelihood + random effects (uniform logits)

( )2j ,N τµ∼θ , ( ) Uniformpitlog ckck ∼=φ



Appendix (Computational Aspects)

Ø There are several packages available for producing statistical analyses

(Bayesian or Frequentist).

Ø Depending on the type of analysis and the technique, the choice of one

package or another could make our life easier (or more miserable).

Ø In general, we can classify the packages in two types:

1. Windows−based packages: (windows = menus)

o Simple to use: Follow the menus

o Little or none freedom: Type of analyses are constrained to the

available routines

o Examples: Statgraphics, Minitab, SPSS, etc…

2. Program−based packages:

o More complicated to use: Need to write your own code (not from

scratch, there are usually lots of available commands)

o Much freedom: Type of analyses are open to the imagination or

needs of the researcher

o Examples: Splus, R, Matlab, WinBugs, etc…

Ø For descriptive statistics (exploratory data analysis) we recommend the use

of a windows-based package



Ø For a more complete inferential analysis (probability model involved) we

recommend a windows-based package, if the routine is available, otherwise

we will require a program-based package.

Ø For a Bayesian analysis we will necessarily require a program-based

package, for example WinBugs.

Ø R (OR SPLUS) ↔ WINBUGS:

o These two packages share the same syntax, although the type of

analysis they produce is different

o Both packages are of free access

o Given to the flexibility of making your own code, R has become very

popular among applied statisticians. Nowadays there are plenty of “R

packages” (routines) freely available for most statistical applications

(Frequentist or Bayesian)

o Some Bayesian books provide code in WinBugs for doing their

examples. Unfortunately our reference book (Spiegelhalter, et al. 2004)

does not do it. However, another book that does provide the code is:

§ Congdon, P. (2001). Bayesian Statistical Modelling, Wiley:

Chichester. (ftp://www.wiley.co.uk/pub/books/congdon)



#R commands for some graphs #An electronic version of this file can be found at #http://allman.rhon.itam.mx/~lnieto/index_archivos/CommandsR.txt #Reading data sets a331<-read.table("http://allman.rhon.itam.mx/~lnieto/index_archivos/A331a.txt",row.names=1) #Assigning names to the columns (variables) dimnames(a331)[[2]]<-c("Type","Sex","Age","TimeCases","TimeControls","CID") #Attach the database to the search path attach(a331) #Barplot for Sex barplot(table(Sex)/dim(a331)[1],names=c("Female","Male"),xlab="Sex", legend=c("0.47","0.53"),col=2:3) title("Barplot for Sex") #Pie chart for Type pie(table(Type),labels=c("Stomach","Bronchus","Colon","Rectum", "Ovary","Breast","Bladder","Kidney","Gall_Bladder","Esophagus", "Reticulum","Prostate","Uterus","Brain","Pancreas","CLL"),col=2:17) title("Piechart for Type (in percentage)") legend(-1.2,0.8,paste(table(Type)),fill=2:17) #Stem and leaf chart for Age stem(Age) #Frequency distribution for Age age.h<-hist(Age,probability=F,plot=F) n<-length(age.h$breaks) age.h1<-cbind(age.h$breaks[1:(n-1)],age.h$breaks[2:n],age.h$counts,age.h$counts/100,cumsum(age.h$counts),cumsum(age.h$counts/100)) dimnames(age.h1)[[2]]<-c("lower.l","upper.l","freq","rel.freq","cum.freq","rel.cum.freq") age.h1 #Histogram for Age hist(Age,probability=T) #Kaplan-Meier plots library(survival) plot(survfit(Surv(TimeCases,CID),conf.int=0),xlab="Time (days)",ylab="Survival probability") lines(survfit(Surv(TimeControls,seq(1,1,,dim(a331)[1]))),lty=3) legend(2000,0.9,c("Cases","Controls"),lty=c(1,3)) title("Kaplan-Meir plots for Cases and Controls") #Summary statistics summary(Age) survfit(Surv(TimeCases,CID),conf.int=0.95) survfit(Surv(TimeControls,CID),conf.int=0.95)



#Boxplots for Cases and Controls boxplot(TimeCases,TimeControls,names=c("Cases","Controls")) #Prior for theta u<-seq(0,1,.01) plot(u,dbeta(u,3,3),type="l",xlab="theta",ylab="",ylim=c(0,3.4)) legend(0.6,3.1,"Prior",lty=1) #Prior + Likelihood plot(u,dbeta(u,3,3),type="l",xlab="theta",ylab="",ylim=c(0,3.4)) lines(u,dbeta(u,3,9),lty=3) legend(0.6,3.1,c("Prior","Likelihood"),lty=c(1,3)) #Prior + Likelihood + Posterior plot(u,dbeta(u,3,3),type="l",xlab="theta",ylab="",ylim=c(0,3.4)) lines(u,dbeta(u,3,9),lty=3) lines(u,dbeta(u,5,11),lty=4) legend(0.6,3.1,c("Prior","Likelihood","Posterior"),lty=c(1,3,4)) #Defining new function: Beta-Binomial density dbebin<- function(x, n = 1, a = 1, b = 1) y <- gamma(a + b)/gamma(a)/gamma(b)/gamma(a + b + n) y <- y * choose(n, x) * gamma(a + x) * gamma(b + n - x) y #Plot of conditional & predictive densities (same graph) fx<-cbind(dbinom(0:10,10,0.31),dbebin(0:10,10,5,11)) barplot(t(fx),xlab="Y",col=2:3,beside=T,legend=c("Binomial","Beta-Binomial")) title("Precitive distribution") #Plot of conditional & predictive densities (different graphs) par(mfrow=c(2,1)) barplot(t(fx[,1]),xlab="Y",col=2,ylim=c(0,0.25)) title("Binomial distribution") barplot(t(fx[,2]),xlab="Y",col=3,ylim=c(0,0.25)) title("Beta-Binomial distribution") #Two noninformative prior densities x<-seq(0,1,0.01) n<-length(x) x<-x[-c(1,n)] par(mfrow=c(1,1)) plot(x,dbeta(x,0.5,0.5),type="l",xlab="theta",ylab="Density") lines(x,x/x,lty=3) legend(0.4,3,legend=c("Jeffreys","Uniform"),lty=c(1,3)) title("Noninformative priors") #Informative prior for Tyrosine x<-seq(0,80,.01) plot(x,dnorm(x,39,sqrt(219.47)),type="l",xlab="theta",ylab="Density") title("Informative prior for Tyrosine")

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